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## Homework Statement

Prove that [tex]\sum(-1)^n\frac{n}{p_n}[/tex] converges, where [tex]p_n[/tex] is the nth prime.

## Homework Equations

The sequence [tex]\frac{n}{p_n}[/tex] is definitely not monotone if there exists infinitely many twin primes, since [tex]2n-p_n<0[/tex] for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?